Determine the following limits.

Question

\lim_{x \to 1^{+}} \frac{x^{2} - 5 x + 6}{x - 1}

Accepted Answer

Welcome back, everyone. Find the limit. Limit as x approaches 2 from the right of X2 minus 9 X + 18 divided by X minus 2, and we're given 4 answer choices A says infinity, B negative infinity, C-2, and D2. So let's va the given limit. It says limit as X approaches 2 from the right. In the numerator, we have X2 minus 9 X + 18. In the denominator that's X minus 2. Now what we're going to do is perform a factorization of the numerator. And then we're going to attempt direct substitution. So first of all, if we want to factor out the quadratic polynomial X2 minus 9 x + 18, we can use the AC method as an example, and we're going to end up with a factor form of X minus 3 multiplied by x minus 6. In the denominator we have x minus 2, and now let's attempt direct substitution. So in the numerator we get. 2 from the right minus 3 multiplied by 2 from the right minus 6. And in the denominator, that's 2 from the right minus 2. Let's notice that in the numerator we're not getting any zeros, so we can ignore the superscript. 2 minus 3 gives us -1 and 2 minus 6 gives us -4. In the denominator, we're going to get 2 from the right minus 2. Now 2 from the right. Basically x approaches 2 from the right. This means that X is slightly larger than 2, and if we are subtracting. 2 from a number that is slightly larger than 2, we're going to get a positive number, right? We're going to approach 0 from the right, so 2 minus 2 gives us 0, but since we're subtracting 2 from a number that is slightly greater than 2, this gives us 0 from the right. Now, let's consider the numerator, -1 multiplied by -4 gives us 4. This is a positive number and we're dividing 4 by 0 from the right, which is also a positive number. The ratio of two numbers that are positive numbers is going to be a positive number, and whenever we divide a number by 0, this corresponds to a limit of infinity. Meaning for this problem, the correct answer corresponds to the answer choice a positive infinity. Thank you for watching.

Determine the following limits.

$\lim_{x \to 1^{+}} \frac{x^{2} - 5 x + 6}{x - 1}$

Key Concepts

Limits

Factoring Polynomials

L'Hôpital's Rule

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